# Finite Element Method Questions and Answers – Plane Elasticity – Governing Equations

This set of Finite Element Method Multiple Choice Questions & Answers (MCQs) focuses on “Plane Elasticity – Governing Equations”.

1. In solid mechanics, what does linearized elasticity deal with?
a) Small deformations in linear elastic solids
b) Large deformations in linear elastic solids
c) Large deformations in non-Hookean solids
d) Small deformations in non-Hookean solids

Explanation: The part of solid mechanics that deals with stress and deformation of solid continua is called Elasticity. Linearized elasticity is concerned with small deformations (i.e., strains and displacements that are very small compared to unity) in linear elastic solids or Hookean solids (i.e., obey Hooke’s law).

2. For plane elasticity problems in three dimensions, which option is not responsible for making the solutions independent of one of the dimensions?
a) Geometry
b) Boundary conditions
d) Material

Explanation: Elasticity is the part of solid mechanics that deals with stress and deformation of solid continua. There is a class of problems in elasticity whose solution (i.e., displacements and stresses) is not dependent on one of the coordinates because of their geometry, boundary conditions, and externally applied loads. Such problems are called plane elasticity problems.

3. For a plane strain problem, which strain value is correct if the problem is characterized by the displacement field ux=ux(x,y), uy=uy(x,y) and uz=0?
a) εxy=0
b) εxz=0
c) εyz≠0
d) εxz≠0

Explanation: The plane strain problems are characterized by the displacement field ux=ux(x,y), uy=uy(x,y) and uz=0, where (ux, uy, uz) denote the components of this displacement vector u in the (x, y, z) coordinate system. The displacement field results in the following strain field:
$$\epsilon_{xz}=\epsilon_{yz}=\epsilon_{zz}=0, \epsilon_{xx}=\frac{\partial u_x}{\partial x}, 2\epsilon_{xy}=\frac{\partial u_x}{\partial y}+\frac{\partial u_y}{\partial x}$$ and $$\epsilon_{yy}=\frac{\partial u_y}{\partial y}$$.

4. Underplane strain condition, what is the value of εyy if the problem is characterized by the displacement field ux=2x+3y, uy=5y2, and uz=0?
a) 10y
b) 5y
c) 3
d) 0

Explanation: The plane strain problems are characterized by the displacement field ux=ux(x,y), uy=uy(x,y) and uz=0, where (ux, uy, uz) denote the components of the displacement vector u in the (x, y, z) coordinatesystem. The displacement field results in the following strain field:
$$\epsilon_{xz}=\epsilon_{yz}=\epsilon_{zz}=0, \epsilon_{xx}=\frac{\partial u_x}{\partial x}, 2\epsilon_{xy}=\frac{\partial u_x}{\partial y}+\frac{\partial u_y}{\partial x}$$ and $$\epsilon_{yy}=\frac{\partial u_y}{\partial y}$$
Thus, $$\epsilon_{yy}=\frac{\partial 5y^2}{\partial y}$$
=5*$$\frac{\partial y^2}{\partial y}$$
=5*2y
=10y.

5.For a plane strain problem, the relation between stress and strain components for an orthotropic material is σ=Cε. Which option is the correct structure of the matrix C?
a) $$\begin{pmatrix}\bar{c}_{11} & \bar{c}_{12} & 0 \\\bar{c}_{12} & \bar{c}_{22} & 0 \\0 & 0 & \bar{c}_{66} \end{pmatrix}$$
b) $$\begin{pmatrix}\bar{c}_{11} & 0 & \bar{c}_{13} \\\bar{c}_{12} & \bar{c}_{22} & 0 \\0 & 0 & \bar{c}_{66} \end{pmatrix}$$
c) $$\begin{pmatrix}\bar{c}_{11} & \bar{c}_{12} & 0 \\-\bar{c}_{12} & \bar{c}_{22} & 0 \\0 & 0 & \bar{c}_{66} \end{pmatrix}$$
d) $$\begin{pmatrix}\bar{c}_{11} & -\bar{c}_{12} & 0 \\\bar{c}_{12} & \bar{c}_{22} & 0 \\0 & 0 & \bar{c}_{66} \end{pmatrix}$$

Explanation: For an orthotropic material under plane strain, with principal material axes (x1, x2, x3) coinciding with the (x, y, z) coordinates, the relation between stress and strain components is $$\begin{pmatrix}\sigma_{xx} \\ \sigma_{yy} \\ \sigma_{xy}\end{pmatrix} = \begin{pmatrix}\bar{c}_{11} & \bar{c}_{12} & 0 \\\bar{c}_{12} & \bar{c}_{22} & 0 \\0 & 0 & \bar{c}_{66} \end{pmatrix} = \begin{pmatrix}\varepsilon_{xx} \\ \varepsilon_{yy} \\ 2 \varepsilon_{xy}\end{pmatrix}$$ where C is the elastic stiffness matrix. The state of stress is σxzyz=0 and $$\sigma_{zz}=E_3(\frac{v_{13}}{E_1}\sigma_{xx}+\frac{v_{23}}{E_2}\sigma_{yy})$$.

6. For an orthotropic material, if E and v represent Young’s modulus and the poisons ratio, respectively, then what is the value of v12 if E1=200 Gpa, E2=160 Gpa and v21=0.25?
a) 0.3125
b) 0.05
c) 0.2125
d) 0.3

Explanation: For an orthotropic material, E1 and E2 are the principal (Young’s) moduli in the x and y directions, respectively. The poisons ratio and Young’s moduli are related by the equation
v12=v21$$\frac{E_1}{E_2}$$.
v12=0.25*$$\frac{200}{160}$$
=0.25*1.25
=0.3125.

7. Under plane stress condition in the XYZ Cartesian system, which stress value is correct if a problem is characterized by the stress field σxxxx(x,y), σyyyy(x,y) and σzz=0?
a) σxy=0
b) σyx≠0
c) σzx≠0
d) σyz≠0

Explanation: A state of plane stress in XYZ Cartesian system is defined as one in which the following stress field exists:
σxzyzzz=0, σxx(x,y), σxyxy(x,y) and σyyyy(x,y).
Thus, σxx , σxy and σyy are non-zero stresses. Such a problem in three dimensions can be dealt with as a two-dimensional (plane) problem.

8. For theplane stress problem in XYZ Cartesian system, σxxxx(x,y), σyyyy(x,y) and σzz=0, which option is correct regarding the associated strain field?
a) εxx=0
b) εyx=0
c) εzx=0
d) εyy=0

Explanation: The strain field associated with the given stress field has the form ε=Sσ, where the matrix S is a symmetric matrix, and it is called elastic compliances matrix. In the XYZ Cartesian system, all the strain components except εyz and εzx are non-zero. Thus, εxx≠0, εyy≠0, εzz≠0, εxy≠0, where as εyz=0 and εzx=0.

9. For any two cases of plane elasticity problems, if the constitutive equations are different, then their final equations of motion are also different.
a) True
b) False

Explanation: The equations of motion for plane elasticity problems are given by D*σ+f=ρü in the vector form, where f denotes body force vector, σ is the stress vector, u is displacement vector, D is a matrix of the differential operator, and ρ is the density. Note that the equations of motion of plane stress and plane strain cases differ from each other only on account of the difference in their constitutive equations.

10. In solid mechanics, which option is not a characteristic of a plane stress problem in the XYZ Cartesian system?
a) One dimension is very small compared to the other two dimensions
b) All external loads are coplanar
c) Strain along any one direction is zero
d) Stress along any one direction is zero

Explanation: An example of a plane stress problem is provided by a plate in the XYZ Cartesian system that is thin along the Z-axis. It is acted upon by external loads lying in the xy plane (or parallel to it) that are independent of the Z coordinate. Thus, stresses and strains are observed in all directions except that the stress is zero along the Z-axis.

11. In solid mechanics, what is the correct vector form of the equations of motion for a plane elasticity problem?
a) D*σ+f=ρü
b) D*σ+f=ρu̇
c) D2*σ+f=ρü
d) D*σ+f=ρu

Explanation: For plane elasticity problems, the equations of motion are one of the governing equations. The vector form of equations of motion is D*σ+f=ρü, where f denotes body force vector, σ is the stress vector, u is the displacement vector, D is a matrix of differential operator and ρ is the density.

12. For plane elasticity problems, which type of boundary condition is represented by the equation tx≡σxxnxxyny, where tx is surface traction force and n is direction cosine?
a) Essential boundary condition
b) Natural boundary condition
c) Both Essential and natural boundary conditions
d) Dirichlet boundary condition